Geoscience Reference
In-Depth Information
or, with (6.8),
⎛
⎞
a
+
1
t
⎝
⎠
q
=
K
r
idt
(6.13)
t
0
in which
t
0
is the starting point of the characteristic; and
/
=
/
/
=
/
(ii)
that
dq
dt
(
dq
dh
)(
dh
dt
)
i
(
dx
dt
), which yields the following integral
for the rate of flow
x
q
=
idx
(6.14)
x
0
where again the lower limit
x
0
is the starting point of the characteristic.
(iii)
The equation of the characteristics
x
=
x
(
t
) is obtained by integration of (6.11),
or with (6.13),
⎛
⎞
a
t
τ
⎝
⎠
x
=
(
a
+
1)
K
r
d
τ
id
σ
+
x
0
(6.15)
t
0
t
0
where
are dummy variables of integration.
The integrals presented in (6.13), (6.14) and (6.15) were first derived by Ishihara and
Takasao (1959) in a critical analysis of the unit hydrograph concept. Smith and Woolhiser
(1971) studied overland flow on an infiltrating surface; they obtained numerical solutions
for the kinematic wave formulation with a lateral inflow
i
(
t
) as the difference between
rainfall rate and infiltration rate obtained from numerical solution of Richards's equation.
Parlange
et al.
(1981) and also Giraldez and Woolhiser (1996) considered different cases
of unsteady lateral inflow and infiltration, i.e.
i
τ
and
σ
i
(
t
), and derived analytical solutions.
The runoff resulting from a steady inflow rate, which was first studied by Henderson
and Wooding (1964), is the key to understanding more general situations. This case is
treated next.
=
6.2.2
Steady lateral inflow
When the lateral inflow remains constant with time, there are two phases of hydrologic
interest. The first is the buildup of the flow on a plane that is initially dry in accordance
with Equation (6.3); the second is the subsidence of the flow after the lateral inflow has
ceased and
i
=
0.
Buildup phase: the rising hydrograph
Since
h
=
0 for
t
=
0 according to Equation (6.3), the integral of (6.12) is
simply
h
=
it
(6.16)
